The diagram shows the graph of a function $f$ - Leaving Cert Mathematics - Question Question 1 - 2012
Question Question 1
The diagram shows the graph of a function $f$.
(a) The graph of another function $g$ is a straight line.
$g(-1) = -6$ and $g(3) = 6$.
Draw the graph of $g$ on the ... show full transcript
Worked Solution & Example Answer:The diagram shows the graph of a function $f$ - Leaving Cert Mathematics - Question Question 1 - 2012
Step 1
Use the graphs to find the two values of $x$ for which $g(x) = f(x)$.
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Answer
The values of x where g(x)=f(x) are found by locating the intersection points of the graphs of g and f. From the graph:
At x=0, g(0)=−3 and f(0)=8 (not equal).
At x=5, g(5)=12 and f(5)=12 (equal).
At x=0, g(0)=−3 and f(0)=−6 (not equal).
Therefore, the solutions are:
x=0 or x=5
Step 2
By finding the values of $a, b, p,$ and $q$, use algebra to solve $g(x) = f(x)$.
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Answer
Given:
g(x)=ax+bf(x)=x2+px+q
The x-intercepts of f are −1 and 3, which provides:
Applying Vieta's formulas:
The sum of roots (−1+3) gives p=−2.
The product of roots (−1∗3) gives q=3.
Thus:
f(x)=x2−2x+3
From the values for g at g(−1) and g(3), we can set up:
For x=−1:−a−b=−6.
For x=3:3a+b=6.
Solving these will yield:
−a−b=−6ightarrowa+b=6.
3a+b=6.
Subtracting these equations gives:
ightarrow a = 3 ; b = -6$$
Thus $g(x) = 3x - 6$. Finally, solving the equation, we find that $g(x) = f(x)$ leads to:\n$$3x - 6 = x^2 - 2x + 3$$
Rearranging gives:
$$x^2 - 5x + 9 = 0$$
The solutions confirm the values of $x$ where the two functions are equal.
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